Teen Prayer In 1995, 40% of adolescents stated they prayed daily. A researcher wants to know whether this percentage has become higher since then. He surveys 40 adolescents and finds that 18 pray on a daily basis. Is there enough evidence to support the proportion of adolescents who pray daily has increased at the α = 0.05 level of significance?

Verified step by step guidance

1. Identify the null and alternative hypotheses. The null hypothesis \(H_0\) assumes the proportion of adolescents who pray daily has not increased, so \(H_0: p = 0.40\). The alternative hypothesis \(H_a\) reflects the research question that the proportion has increased, so \(H_a: p > 0.40\).
2. Determine the sample proportion \(\hat{p}\) by dividing the number of adolescents who pray daily by the total surveyed: \(\hat{p} = \frac{18}{40}\).
3. Calculate the test statistic for a one-proportion z-test using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \] where \(p_0 = 0.40\) is the hypothesized population proportion and \(n = 40\) is the sample size.
4. Find the critical value for a right-tailed test at the \(\alpha = 0.05\) significance level using the standard normal distribution (z-distribution). This critical value is the z-score such that the area to the right is 0.05.
5. Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude there is sufficient evidence to support that the proportion of adolescents who pray daily has increased. Otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing for Proportions

Hypothesis testing for proportions involves assessing whether a sample proportion provides enough evidence to support a claim about a population proportion. It starts with a null hypothesis (no change) and an alternative hypothesis (increase), and uses sample data to decide whether to reject the null at a given significance level.

Significance Level (α)

The significance level, denoted by α, is the threshold probability for rejecting the null hypothesis. It represents the risk of a Type I error—incorrectly concluding there is an effect when there isn't one. Commonly set at 0.05, it means there is a 5% chance of a false positive.

Test Statistic and p-value

The test statistic measures how far the sample proportion deviates from the hypothesized proportion under the null hypothesis. The p-value quantifies the probability of observing such a result (or more extreme) if the null is true. Comparing the p-value to α helps decide whether to reject the null hypothesis.